Nonlinear Dynamics

, Volume 88, Issue 4, pp 2491–2501 | Cite as

Evoking complex neuronal networks by stimulating a single neuron

  • Mengjiao Chen
  • Yafeng Wang
  • Hengtong Wang
  • Wei Ren
  • Xingang Wang
Original Paper


The responses of electrically coupled neuronal network to external stimulus injected on a single neuron are investigated. Stimulating the largest-degree neuron in the network, it is found that as the intensity of the stimulus increases, the network will be transiting from the resting to firing states and then restoring to the resting state, thereby showing a bounded firing region in the parameter space. Furthermore, it is found that as the coupling strength among the neurons decreases, the firing region is gradually expanded and, at the weak couplings, it could be separated into several disconnected subregions. By a simplified network model, we conduct a detailed analysis on the bifurcation diagram of the network dynamics in the two-dimensional parameter space spanned by stimulating intensity and coupling strength, and, by introducing a new coefficient named effective stimulus, explore the underlying mechanisms for the modified firing region. It is revealed that the coupling strength and stimulating intensity are equally important in evoking the network, but with different mechanisms. Specifically, the effective stimuli are shifted up globally by increasing the stimulating intensity, while are drawn closer by increasing the coupling strength. The dynamical responses of small-world and random complex networks to external stimulus are also investigated, which confirm the generality of the observed phenomena. The findings shed new lights on the collective behaviors of complex neuronal networks and might help our understandings on the recent experimental results.


Complex neuronal network Coupled oscillators Excitability Bifurcations 



This work was supported by the National Natural Science Foundation of China under the Grant No. 11375109, and by the Fundamental Research Funds for the Central Universities under the Grant Nos. GK201601001 and GK20150-3027.


  1. 1.
    Kandel, E.R., Schwartz, J.H., Jessell, T.M., Siegelbaum, S.A., Hudspeth, A.J.: Principles of Neural Science. McGraw-Hill Medical, New York (2012)Google Scholar
  2. 2.
    Izhikevich, E.M.: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press, Cambridge (2007)Google Scholar
  3. 3.
    Reti, I.: Brain Stimulation: Methodologies and Interventions. Wiley, Hoboken (2015)CrossRefGoogle Scholar
  4. 4.
    Brecht, M., Schneider, M., Sakmann, B., Margrie, T.W.: Whisker movements evoked by stimulation of single pyramidal cells in rat motor cortex. Nature 427, 704–710 (2004)CrossRefGoogle Scholar
  5. 5.
    de la Prida, L.M., Huberfeld, G., Cohen, I., Miles, R.: Threshold behavior in the initiation of hippocampal population bursts. Neuron 49(1), 131–142 (2006)CrossRefGoogle Scholar
  6. 6.
    Houweling, A.R., Brecht, M.: Behavioural report of single neuron stimulation in somatosensory cortex. Nature 451(7174), 65–68 (2008)CrossRefGoogle Scholar
  7. 7.
    Morgan, R.J., Soltesz, I.: Nonrandom connectivity of the epileptic dentate gyrus predicts a major role for neuronal hubs in seizures. Proc. Natl. Acad. Sci. USA 105(16), 6179–6184 (2008)CrossRefGoogle Scholar
  8. 8.
    Li, C.Y., Poo, M.M., Dan, Y.: Burst spiking of a single cortical neuron modifies global brain state. Science 324(5927), 643–646 (2009)CrossRefGoogle Scholar
  9. 9.
    Bonifazi, P., Goldin, M., Picardo, M.A., Jorquera, I., Cattani, A., Bianconi, G., Represa, A., Ben-Ari, Y., Cossart, R.: GABAergic hub neurons orchestrate synchrony in developing hippocampal networks. Science 326(5958), 1419–1424 (2009)CrossRefGoogle Scholar
  10. 10.
    Luccioli, S., Ben-Jacob, E., Barzilai, A., Bonifazi, P., Torcini, A.: Clique of functional hubs orchestrates population bursts in developmentally regulated neural networks. PLoS Comput. Biol. 10(9), e1003823 (2014)CrossRefGoogle Scholar
  11. 11.
    Lau, P.M., Bi, G.Q.: Synaptic mechanisms of persistent reverberatory activity in neuronal networks. Proc. Natl. Acad. Sci. USA 102(29), 10333–10338 (2005)CrossRefGoogle Scholar
  12. 12.
    Zhao, Z., Jia, B., Gu, H.: Bifurcations and enhancement of neuronal firing induced by negative feedback. Nonlinear Dyn. 86(3), 1549–1560 (2016)CrossRefGoogle Scholar
  13. 13.
    Fontanini, A., Katz, D.B.: Behavioral states, network states, and sensory response variability. J. Neurophysiol. 100(3), 1160–1168 (2008)CrossRefGoogle Scholar
  14. 14.
    Landisman, C.E., Connors, B.W.: Long-term modulation of electrical synapses in the mammalian thalamus. Science 310(5755), 1809–1813 (2005)CrossRefGoogle Scholar
  15. 15.
    Kothmann, W.W., Trexler, E.B., Whitaker, C.M., Li, W., Massey, S.C., O’Brien, J.: Nonsynaptic NMDA receptors mediate activity-dependent plasticity of gap junctional coupling in the AII amacrine cell network. J. Neurosci. 32(20), 6747–6759 (2012)CrossRefGoogle Scholar
  16. 16.
    Connors, B.W., Long, M.A.: Electrical synapses in the mammalian brain. Annu. Rev. Neurosci. 27, 393–418 (2004)CrossRefGoogle Scholar
  17. 17.
    Sun, A., Lü, L., Li, C.: Synchronization of an uncertain small-world neuronal network based on modified sliding mode control technique. Nonlinear Dyn. 82(4), 1905–1912 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Sun, H., Zhang, H., Wu, J.: Correlated scale-free network with community: Modeling and transportation dynamics. Nonlinear Dyn. 69(4), 20972104 (2012)Google Scholar
  19. 19.
    Yang, X.L., Hu, L.P., Sun, Z.K: How time-delayed coupling influences differential feedback control of bursting synchronization in modular neuronal network. Nonlinear Dyn. 86(3), 1797–1806 (2016)Google Scholar
  20. 20.
    Zhang, J., Huang, S., Pang, S., Wang, M., Gao, S.: Optimizing calculations of coupling matrix in Hindmarsh–Rose neural network. Nonlinear Dyn. 84(3), 1303–1310 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sporns, O.: Networks of the Brain. MIT Press, Cambridge (2010)MATHGoogle Scholar
  22. 22.
    Eguíluz, V.M., Chialvo, D.R., Cecchi, G.A., Baliki, M., Apkarian, A.V.: Scale-free brain functional networks. Phys. Rev. Lett. 94, 018102 (2005)CrossRefGoogle Scholar
  23. 23.
    Sporns, O., Honey, C.J., Kötter, R.: Identification and classification of hubs in brain networks. PLoS One 2(10), e1049 (2007)CrossRefGoogle Scholar
  24. 24.
    Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., Alon, U.: Network motifs: simple building blocks of complex networks. Science 298(5594), 824–827 (2002)CrossRefGoogle Scholar
  25. 25.
    Reigl, M., Alon, U., Chklovskii, D.B.: Search for computational modules in the C. elegans brain. BMC Biol. 2, 25 (2004)CrossRefGoogle Scholar
  26. 26.
    Sporns, O., Kötter, R.: Motifs in brain networks. PLoS Biol. 2(11), e369 (2004)CrossRefGoogle Scholar
  27. 27.
    Felleman, D.J., Van Essen, D.C.: Distributed hierarchical processing in the primate cerebral cortex. Cereb. Cortex 1(1), 1–14 (1991)CrossRefGoogle Scholar
  28. 28.
    Hilgetag, C.C., Burns, G.A., O’Neill, M.A., Scannell, J.W., Young, M.P.: Hierarchical organization of macaque and cat cortical sensory systems explored with a novel network processor. Philos. Trans. R. Soc. Lond. B Biol. Sci. 355(1393), 71–89 (2000)CrossRefGoogle Scholar
  29. 29.
    Buono, P.L., Golubitsky, M.: Models of central pattern generators for quadruped locomotion. I. Primary gaits. J. math. Biol. 42(4), 291–326 (2001)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Söhl, G., Maxeiner, S., Willecke, K.: Expression and functions of neuronal gap junctions. Nat. Rev. Neurosci. 6, 191 (2005)CrossRefGoogle Scholar
  31. 31.
    Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35(1), 193–213 (1981)CrossRefGoogle Scholar
  32. 32.
    Hodgkin, A.L.: The local electric changes associated with repetitive action in a non-medullated axon. J. Physiol. (London) 107(2), 165–181 (1948)Google Scholar
  33. 33.
    Gutkin, B.S., Ermentrout, G.B.: Dynamics of membrane excitability determine interspike interval variability: a link between spike generation mechanisms and cortical spike train statistics. Neural Comput. 10(5), 1047–1065 (1998)CrossRefGoogle Scholar
  34. 34.
    Agmon-Snir, H., Carr, C.E., Rinzel, J.: The role of dendrites in auditory coincidence detection. Nature 393(6682), 268–272 (1998)CrossRefGoogle Scholar
  35. 35.
    Stiefel, K.M., Englitz, B., Sejnowski, T.J.: Origin of intrinsic irregular firing in cortical interneurons. Proc. Natl. Acad. Sci. USA 110(19), 7886–7891 (2013)CrossRefMATHGoogle Scholar
  36. 36.
    Tsumoto, K., Kitajima, H., Yoshinaga, T., Aihara, K., Kawakami, H.: Bifurcations in Morris-Lecar neuron model. Neurocomputing 69(4–6), 293–316 (2006)CrossRefGoogle Scholar
  37. 37.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Bennett, M.V.L., Zukin, R.S.: Electrical coupling and neuronal synchronization in the mammalian brain. Neuron 41(4), 495–511 (2004)CrossRefGoogle Scholar
  39. 39.
    Erdös, P., Rényi, A.: On random graphs. Publ. Math. Debrecen 6, 290–297 (1959)MathSciNetMATHGoogle Scholar
  40. 40.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’ networks. Nature 393(6684), 440–442 (1998)CrossRefGoogle Scholar
  41. 41.
    Koukkari, W.L., Sothern, R.B.: Introducing Biological Rhythms. Springer, New York (2006)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • Mengjiao Chen
    • 1
  • Yafeng Wang
    • 1
  • Hengtong Wang
    • 1
  • Wei Ren
    • 2
  • Xingang Wang
    • 1
  1. 1.School of Physics and Information TechnologyShaanxi Normal UniversityXi’anChina
  2. 2.College of Life Science, Key Laboratory of MOE for Modern Teaching TechnologyShaanxi Normal UniversityXi’anChina

Personalised recommendations